Download (normal) Zeeman Effect with Spin Spin

(normal)
Zeeman Effect with Spin
In strong external magnetic fields B, orbital and spin magnetic moments are both
coupled to B (and not to each other): In this case, z-components of the moments are
additive.
No magnetic field
With magnetic field, no spin
With magnetic field and spin
l=1, ml=+1, ms=+1/2
l=1, ml=0, ms=+1/2
l=1, ml=+1, ms=-1/2 and
l=1, ml=-1, ms=+1/2
l=1, ml=0, ms=-1/2
n=2, l=1
l=1, ml=-1, ms=-1/2
l=0, ml=0, ms=+1/2
n=1, l=0
l=0, ml=0, ms=-1/2
(note: no spin-orbit coupling in this diagram. We will later learn that some transitions are forbidden)
Spin-Orbit Interaction
In the absence of an external magnetic field (or a weak field), orbital and spin
magnetic moments interact.
From the perspective of the electron, the (positively
charged, Ze+) nucleus orbits around it. This creates a
magnetic moment at the site of the electron.
The electron energy is higher (less stable), if spin S is aligned with B.
The electron energy is lower (more stable), if spin S is aligned opposite to B.
Spin-Orbit Vector Addition
When spin-orbit coupling is in effect, neither
L nor S are angular momentum conserved;
only the total angular momentum J=L+S is
conserved. This replaces the set of quantum
numbers
(l, ml, s, ms)
with a new set.
(l, s, j, mj)
l=2, s=1/2 ! j=5/2 and j=3/2
Total angular momentum vector:
l=2, j=5/2, mj=3/2
Quantum numbers:
Spectroscopic Notation (Term Symbols)
General Form:
l : orbital angular momentum quantum number.
s: spin quantum number.
j: total angular momentum quantum number.
2s+1 : “Spin Multiplicity” " total number of spin states
[e.g. 2s+1=2 (up and down) for s=1/2].
Example: hydrogen atom (lowest states):
(note: In this example, spin multiplicity is always 2 because hydrogen has only one electron.
Total angular momentum quantum number j is either l+1/2 or l-1/2).
Dirac Relativistic Hydrogen Atom
The Bohr/Schrödinger result for hydrogen
Bohr/Schrödinger
Dirac
is
based
on
non-relativistic
(Newtonian) mechanics.
The (special-) relativistic equivalent to
the Schrödinger equation is the Dirac
equation. The Dirac equation, naturally
explains spin (and spin-orbit coupling),
giving the following energy equation of
hydrogen.
(Fine structure constant)
Shift of Dirac levels relative to Bohr/Schrödinger is
exaggerated by a factor of (1/#2).